A hydraulic model for the catchment-stream problem

Journal of Hydrology 3 (1965) 268-282; © North-Holland Publishing Co., Amsterdam N o t to be reproduced by photoprint or microfilm without written permission from the publisher

A HYDRAUIJC

MODEL

FOR THE CATCHMENT-STREAM PROBLEM II. Numerical solutions R. A. WOODING CSIRO, Div&ion of Plant Industry, Canberra, A.C.T., Australia

Abstract: With the aid of kinematic-wave theory described in Part I for flow over a catchment and along a stream, the predicted form of stream hydrograph is calculated numerically, assuming that the rainfall is of constant intensity and of finite duration. The influence of the parameter 2, a measure of the ratio of stream "response time" to catchment "response time", becomes significant for ~ : 0(1); the general effect of large ~, is to delay the arrival time of a flood peak and to reduce its amplitude. For 2 small, the catchment discharge function is a good guide to the form of the stream hydrograph. With finite infiltration, the duration of catchment discharge is finite.

1. Numerical solution of the differential equations I n P a r t 11), a description has been given o f a simplified r u n o f f m o d e l c o m p o s e d o f a V - s h a p e d c a t c h m e n t d r a i n i n g into a line stream situated at the apex o f the V, the two halves o f the c a t c h m e n t being a s s u m e d plane. A n a l y t i cal s o l u t i o n s for the s t r e a m outflow are o b t a i n e d on the a s s u m p t i o n s t h a t p o w e r - l a w types o f d e p t h - d i s c h a r g e relation a p p l y for flow over the catchm e n t a n d in the stream, a n d t h a t the rainfall is u n i f o r m l y distributed, c o n stant, a n d o f finite d u r a t i o n . T o generalize the p r o b l e m to cases where the rainfall varies a r b i t r a r i l y with time, it is necessary to r e t u r n to the differential e q u a t i o n s governing the c a t c h m e n t a n d s t r e a m flows, a n d to c o n s i d e r their s o l u t i o n by n u m e r i c a l or g r a p h i c a l means. N u m e r i c a l solutions u p o n a digital c o m p u t e r are generally the m o r e accurate a n d convenient. This m e t h o d is also o f assistance in the r a p i d p r o d u c t i o n o f solutions for which analytical results are a l r e a d y available. Here it will be c o n v e n i e n t to c a r r y over the n o t a t i o n o f P a r t I. D i m e n s i o n less variables are defined for one o f three alternative schemes, d e p e n d i n g u p o n w h e t h e r (i) the rainfall intensity or (ii) t o t a l rainfall or (iii) rainfall dur a t i o n is t a k e n as an a p p r o p r i a t e scale factor. 268

THE CATCHMENT-STREAM PROBLEM

269

The characteristic equations for the catchment are dh --

dt

dx

--

=

v -

f,

= nh"- 1,

(1)

(2)

dt and the discharge per unit catchment width is

q = h"lx= .

(3)

Here t is the time, x is the distance measured down the catchment, v - f is the excess rainfall, assumed spatially uniform, and h is the depth of water at distance x on the catchment. For the stream, d ( 2 H ) = q]~=l, d

(;tx) = N H " - i,

(4)

(5)

and the discharge at the outlet point is

Q = HNIx=I,

(6)

where X is distance downstream, H is stream stage and 2 is a dimensionless parameter relating the "time constants" of stream and catchment. The above quantities must obey the restrictions that S(v - f ) dt is bounded for any finite range of integration over t, and that h and H are non-negative for all t. The problem is specified completely by assuming that v - f given as a function of time for t >/0, that the spatial distributions of h and H are known at t = 0, and that the upstream inflow (at X = 0 ) is known as a function of time. It is assumed that the depth h is zero at x = 0, the top of the catchment, an approximate boundary condition which is adequate where all depths are small2). In the (x, t)-plane, then, the quantity h is known along the positive semi-axes (x= 0, t/> 0) and (x >/0, t = 0), while in the (X, t)-plane the quantity H is determined in a similar way. Since only ordinary differential equations are involved in (l), (2), (4) and (5), these may be solved by forward integration with starting points on the axes, using one of the standard numerical procedures. Eqs. (1) and (2) are solved simultaneously along successive, suitably spaced characteristics in the (x, t)-plane. Values of h and t at which each characteristic intersects the line x = 1 are found by interpolation and then used to define the function q]x= 1 on the right hand side of eq. (4), using a further interpolation procedure to determine values spaced equally in time. Finally, eqs. (4) and (5) are solved

270

R.A. WOODING

simultaneously, the stream outflow function being found from values of H andtatX=l. Since (4) and (5) are also solved along successive characteristics, it is apparent that the numerical values of q must be stored for repeated use. This complicates the use of a fixed-error method of numerical integration based upon an adjustable time increment since frequent interpolations are necessary. In the present case, a fixed time increment has been chosen, using a simple Runge K u t t a integration routine and keeping the step length small enough to ensure that errors are within acceptable limits. A large number of integration steps are involved - of order 104 in a typical case - and the most efficient approach results if 2 H and 2nX are chosen as dependent variables in place of H and X. Stream outflow functions corresponding to different 2i i = 1, 2 . . . . are then built up by finding the points of intersection of each characteristic with the lines 2~X=2~; thus an indefinite number of values of 2 can be treated in "one pass". In a typical case, using a C D C 3600 Computer, the outflow functions corresponding to eight values of 2 are obtained in about one minute. If an automatic plotter is used to present the output in graphical form, a further period of 1 or 2 minutes is required. This approach is particularly useful if the model is being "calibrated" to represent, as nearly as possible, the outflow from a given catchment under an arbitrary rainfall. Generally, the rainfall time scale and the appropriate value of 2 cannot be calculated directly since the parameters of roughness, etc. are unknown. However, it is a fairly straightforward matter to use suitable rainfall data and, by adjustment of 2 and the time scale, to match the calculated output to the measured stream outflow.

2. Effects of varying rainfall Some general properties of the stream outflow function Q can be deduced by varying the parameters of the steady rainfall of finite duration ("square wave"). Here the numerical solutions are examined in some detail for three cases. It is assumed that infiltration effects are negligible and, as in Part I, that the exponents in the depth-discharge eqs. (3) and (6) are n = 2 and N= 3/2 respectively. 2 . 1 . TOTAL RAINFALL h 0 CONSTANT

In terms of the dimensionless variables defined in (ii) of Part I, the area under the rainfall function is unity, while the shape of the square wave varies as the duration to is varied. Catchment outflow curves are given in Fig. 1(a) for five values of to. (These are similar to curves given previously by Hender-

271

THE CATCHMENT-STREAM PROBLEM

son and Wooding 2).) The notable property of these catchment outflow curves is the behaviour of the flat-topped peak. For t o < 1, the peak value persists over the (dimensionless) time interval (t o, ½to +½), which shrinks to zero as t o ~ l , equal to the time required for the catchment flow to reach a steady state. At the same time the peak outflow rate remains constant at unity (or 1.0

.2

CATCHMENT OUTFLOW

q 0'5

to=4

(a)

0

]

2

3

4

Fig. 1. Calculated forms of the catchment and stream discharge functions q and Q for a steady rainfall o f finite duration to, with total rainfall h0 constant. (a) Catchment outflow, (b) - (e) stream outflow with 2 =- 0.2, 0.5, 1 and 2 respectively. -2

1.0

•5

I

x ='2

Q 0.5

to~4

(b)

o

!

2

3

4

272

R . A . WOODING

~ho2 in dimensional units). When to > 1, the peak is flat-topped over the interval (to~, to), while the peak outflow rate is equal to l/to (or Lho/t o in dimensional units), corresponding to a steady-state condition of flow over the catchment. For comparison, it should be observed that a purely linear theory would show no variation in the shape of the outflow curve. The resultant stream outflow curves for 2=0.2, 0.5, 1 and 2 are given in Figs. l(b) to (e) and indicate, in a general way, the manner in which the

1"0 ~.5

X-.5

Q

I

2

0"5

to:4

(c)

0

I

2

3

4

1"0

I

Xol

°2

•

Q 0"5

(d)

I

2

3

4

273

THE CATCHMENT-STREAM PROBLEM

~.-2

0.5 Q •2

(e)

0

1

2

.5

1

3

4

hydrograph is influenced by various stream "response times". It is evident that, as 2-,0, the stream outflow tends to the same form as the catchment outflow. However, with increasing 2 the width of the flat-topped peak progressively decreases, disappearing completely at 2 = 0.5 for to not too large; for further increase in 2 the peak discharge rate decreases fairly rapidly. At the same time, the rising portion of the stream hydrograph develops an S-shape not observable in the curve of catchment outflow. These points are indicated more precisely in Fig. 2(a) and (b). Fig. 2(a) gives the time, ?, as a function of t 0, at which the catchment discharge rate or stream discharge rate has its maximum value (~ or 0 respectively). The shaded portions indicate flat-topped peaks. For 2 > 0 there always exists a range of values of t o for which the stream-outflow peak is not flat-topped. For 2--0.5 and to-*0, the peak time tends asymptotically to the arrival time of the rear edge of the flat catchment peak, i.e., this time coincides exactly with the time required by the stream to reach a state of steady flow; the curve )°=0.5 separates two regions of fundamentally different properties. Fig. 2(b) shows the variation of peak discharge rate with to for the above values of 2. For 2 <0.5 the peak stream discharge rate is strongly controlled by the catchment, since the peak flow generally corresponds to the steadystate condition of flow in the stream. By contrast, when 2 >0.5 the stream discharge peak 0 tends to the upper limit 1 - ½ log 22

(7)

as to~0. This important result, which is demonstrated in Fig. 2(c), serves to illustrate the extent to which stream-bed treatment might be expected to control flood peaks.

274

R.A.

WOODING '

-

,

4

X o2

d J ~,,s f

2

I

?

¢

.5

.2

I

(a)

1

I

I

•2

I I III

"5

I I

I

I

I

2

5

to

Fig. 2 (a). Peak-arrival times t and (b) peak-discharge rates 4 and (~ from catchment (full lines) and stream (broken lines) versus rainfall duration to, for a steady rainfall with total h0 constant. Shaded areas indicate flat-topped peaks. Dimensionless variables are defined in (ii) of Part Ix). (C) Values of ~ from the same data plotted versus 2; the broken line indicates the limit as t 0 - + 0. CATCHMENT I-0

-5 "4

"3

(b)

I •2

i

i

w

t

I

.5

I

2

to

1

I

I $

THE CATCHMENT-STREAM

275

PROBLEM

t o • "2

1.0

"6

a

\\

2

I

-4 --

N\

"2

(c)

r I I

I

I

~ I r

I

I

"05

.1

"2

"5

I

2

•02

2.2.

RAINFALL DURATION

tO C O N S T A N T

Here the appropriate dimensionless variables are defined in (iii) of Part 1. The length of the rainfall " s q u a r e w a v e " becomes unity, while the dimensionless rainfall intensity v o m a y be varied. W h e n v 0 = 1 the time to achieve a steady state in flow over the catchment is equal to unity. Since a purely linear theory would predict a constant arrival time for the flood peak, with a peak discharge rate linearly p r o p o r t i o n a l to rainfall, the present study indicates h o w the non-linear model differs in these f u n d a m e n t a l properties. Fig. 3(a) shows the arrival time of catchment- and stream-flow peak discharges, while Fig. 3(b) gives the corresponding peak discharge rates. It is significant that, as Vo is increased, the peak arrival times t are progressively advanced; the c a t c h m e n t p e a k is flat-topped over the interval (1, ~am±~ U1,,O la) for 0 < v 0 < 1 and over the interval (Vo ~, 1) for v > 1

276

R.A. WOODING

For 2 > 0 and vo sufficiently small, the stream outflow peak is generally flat-topped since the catchment discharge remains constant while the stream flow achieves a steady state. (A steady state does not exist in the flow over the catchment, however.) In this region the peak discharge rates of both catchment and stream vary as v~ (cf. Fig. 3(b)). At sufficiently large values of Vo, both catchment and stream rapidly achieve a steady state and the outflow peak is again flat-topped, but with discharge rate now proportional to %. Finally, for 2 > 1 there exists a finite range of values of vo (containing the value Vo= l) for which the stream flow does n o t achieve a steady state, and for which the peak discharge rate varies approximately as v~"5. 2.3. RAINFALLINTENSITY1)0 CONSTANT With the dimensionless variables defined in (i) of Part I, the intensity of the rainfall "square wave" is unity, while the time to achieve a steady state in flow over the catchment is constant and equal to unity, For 0 < t o < 1, the catchment discharge peak is fiat-topped over the interval (t o, 1to+-12to 1) (cf. Part I and Fig. 4(a)) while the peak discharge rate is pro-

% % --

?

% •~,%

2

CATCHMENT

(a)

.5

I •2

1

1

l

I

,

"5

1

2

% Fig. 3(a). Peak-arrival times i and (b) peak-discharge rates 4 and ~ from catchment (full lines) and stream (broken lines) versus rainfall intensity v0, for a steady rainfall with duration to constant. Shaded areas indicate fiat-topped peaks. Dimensionless variables are defined in (iii) of Part 11).

277

THE CATCHMENT-STREAM PROBLEM

2

~

# # !

/

-

Ill

"~ l

~'i !l

°~

ff /

///

~

Ii! /

I

#

.05

(b)

I •2

I

I

I

/ # ,,

// I

!

1

I

/ I

I

I

I

.5

1

2

vo

portional to t~ (Fig. 4(b)); when to > 1 the peak is fiat-topped over the interval (1, to) with peak discharge rate independent of to. As in cases (a) and (b) above, the properties of the stream outflow function for 2 > 0 are strongly influenced by the nature of the catchment discharge function. The notable feature of this case, illustrated in Fig. 4(a), is that the arrival time 7 of the stream discharge peak has a minimum value at a finite value of t o. At first 7 decreases with increasing t o as the stream rate-of-rise increases; however, 7 ultimately increases as the "centre-of-gravity" of the catchment discharge becomes more retarded in time. 3. Rainfall function of other forms

The results of section 2, in which are discussed the effects upon outflow of maintaining constant (a) the total rainfall, (b) the duration and (c) the intensity, while varying other parameters of a "square-wave" rainfall function, may be applied qualitatively to functions of other forms. Particular differences should be noted. Some recorded rainfalls may be represented approximately by triangular functions3), and for such cases flat-topped hy-

278

R . A . WOODING

?

I

(a)

F •2

"5

1

2

1:0

Fig. 4(a). Peak-arrival times ~ and (b) peak-discharge rates 4 and 0~ from catchment (full lines) and stream (broken lines) versus rainfall duration to, for a steady rainfall with intensity v0 constant. Shaded areas indicate flat-topped peaks. Dimensionless variables are defined in (i) of Part I1). CATCHMENT

1.0

//.s / / / / / / ' /,

/ /I

.i// ,, U I .i i //i

I/ I

.2

•I

(b)

--

.05 .2

/

/

/

/

/

/

/

I

I

I

•$

I

2

tO

I

l

THE CATCHMENT-STREAM PROBLEM

279

drographs will not be encountered. However, in the cases treated above the catchment outflow function is approximately triangular over part of each parameter range; thus the differences to be expected are not as great as would at first appear. With a triangular rainfall function the treatment would closely follow that given above, noting that one further parameter, specifying the triangle shape, would have to be introduced. 4. Modifications due to infiltration

In section 2, the dominant effect exerted by the catchment discharge in determining the form of the stream hydrograph has been clearly indicated for all cases except those in which the parameter )4 is unusually large. When a finite amount of infiltration is present, the effects upon stream outflow may be inferred qualitatively from the calculated effects upon the catchment discharge. Hence only the latter are considered here. For consistency with the results of Part I, section 4, it is appropriate to return to the original (dimensional) variables, with two-dimensional flow of depth h over a catchment of length L, x being measured from the top of the catchment slope. For the simple case of rainfall starting at time t = 0, of constant intensity vo, with infiltration at the steady rate fo ( < Vo), the flow over the catchment reaches a steady state after a time t~ given by (Vo - f o ) t~ = {(Vo - f o ) L/a} ~

(8)

where c~ is a parameter determined by the slope and roughness (cf. Part 1, section 2), and it is assumed that the exponent n in the depth-discharge power law has the Horton value of 2. Now, let the steady rainfall have duration to. Since infiltration continues after rainfall has ceased, the depth h becomes zero at a finite time, ti say, greater than t o. The solution for the catchment depth h at the discharge point x = L is summarized below. Case I (to >- ts) Phase Phase Phase Phase

l: 2: 3: 4:

0
h=(vo-fo)t. h = (v o - f o ) t s . h = (v o - f o ) ~ { V o ( t - to) z + (Vo -fo)t~z}½- V o ( t - to). h--O.

(9)

Here ts is given in (8), while

t, = to + t~(Vo - f o ) / (Vofo) ~ Case IIa ((fo/Vo) "~ ts < to < ts)

(lo)

280

Phase Phase Phase Phase

R . A . WOODING

1: 2: 3: 4:

0
o,

t o < t < t p, tp
h=(vo-fo)t. h = v o t o - f o t. h=(vo-fo)~{Vo(t-to)Z-t-(vo-fo)t~)~-Vo(t-to). h = O.

(11)

T h e d e f i n i t i o n o f t i i n (10) is r e t a i n e d , w h i l e tp is g i v e n b y

fotp = rot o -- (v o -- fo)½ (Vot~ --fot2s) ~.

h:o

1.oI

(12)

.2

0"5

(a) 0

l

2

3

4

5

t

i-o

fo o o

q

to= I 0"5 .4

"8

(b) 0

I

2

3

I

J

4

5

t

Fig. 5. Effects of infiltration fo upon the catchment discharge q due to a steady rainfall with intensity vo given, for rainfall durations of (a) to -- 2, (b) to = 1 and (c) to ~ 0~5. Dimensionless variables are defined in (i) of Part 11).

281

THE CATCHMENT-STREAM PROBLEM

to •

"5

0"5

fo-O

(c)

1

2

3

J4

t

Case l i b (0 < to < (fo / Vo)~ts) Phase 1: 0 < t < t o, Phase 2: t o < t < t i, Phase 3: ti < t,

h=(vo-fo)t. h=voto-fot. h = O.

For this case, ti = (vo/fo)to.

(13)

(14)

F r o m (12) and (14) it will be seen that the condition t o = (fo/Vo)~ts corresponds to t i = t o. It will be noted that the "tail" of the catchment discharge function is now of finite length. Only in case I is the catchment discharge function truly flattopped. In case II the level increases up to t = t o and then begins to decrease; case I I a involves an outflow function with three finite segments, while case l i b involves only two segments forming a curvilinear triangle. The transformation of these equations to dimensionless variables is carried out as described in Part I, section 4, noting that the scale factors are based upon total rather than net, rainfall. Figs. 5(a) to (c) indicate the effects, upon the catchment discharge q = h z, of progressively increasing amounts of infiltration for a rainfall of given intensity. Here use is made of the dimensionless variables defined in (i) of Part I. In Fig. 5(a), all curves belong to Case I (eqs. (9) above); Fig. 5(b) illustrates the marginal situation (to = ts in dimensional units); in Fig. 5(c) the curves f o r f o =-0.1 and 0.2 belong to Case lla while those f o r f o =0.4 and 0.8 belong to Case lib. These results lead to certain useful conclusions concerning the total quantity of water discharged from a catchment. The proportion of water discharged as surface runoff is equal to the ratio of the area under the q - t curve to the dimensionless total rainfall to. If the ratio is plotted as a function of the dimensionless precipitation fo, the resultant curve is concave upwards (Fig. 6). This is an obvious consequence of the fact that infiltration continues

282

R.A. WOODING

\\ \\ \ =

.5

I'0

fo Fig. 6. Curves showing the proportion of total rainfall discharged as surface runoff as a function of the infiltration rate. Dimensionless variables defined in (i) of Part 11) are used. The broken line indicates the proportion of water (1 --f0) which would be discharged during a continuous, steady rainfall. while there is still surface water present, which m a y be for a m u c h longer p e r i o d t h a n the rainfall d u r a t i o n . T h e t a s k o f estimating a final infiltration rate is c o m p l i c a t e d b y the fact t h a t infiltration m a y be c o n t i n u i n g over the lower p a r t o f a c a t c h m e n t after all surface water has d i s a p p e a r e d f r o m the u p p e r part. F o r this r e a s o n the curves o f Fig. 6 c o u l d p r o v i d e an estimate o f fo in p r a c t i c a l cases where values o f vo, to a n d ts can be d e t e r m i n e d with sufficient accuracy.

References l) R. A. Wooding, Journal of Hydrology 3 (1965) 254 2) F. M. Henderson, and R. A. Wooding, Journal of Geophysical Research 69 (1964) 1531 3) E. E. Dawson, Proceedings of the Institute of Civil Engineers 10 (1958) 453